Burton's three scoring catches broke the multiple-touchdown streak that teammate Dicky Lyons Jr. set in the season's first three games.
His third one Saturday, a 25-yarder from Andre Woodson, extended Kentucky's lead to 8 points late in the fourth quarter and moved Woodson into a tie for third on the Wildcats' all-time list with 27 scoring passes this year.
Monroe quarterback Kinsmon Lancaster led the Warhawks downfield and scored on a one-yard sneak, cutting the lead to 2 with a minute left. But Lancaster couldn't reach the end zone on the 2-point conversion try.
The Wildcats recovered an onside kick attempt and ran the clock down to four seconds, but then fumbled to give the Warhawks the ball back at midfield for one last play. Lancaster completed only a short pass before time expired.
Kentucky (7-4, 4-3 Southeastern Conference) was already eligible for its first bowl game since 1999 and can clinch second place in the SEC East by winning its regular season finale at No. 22 Tennessee. The Wildcats have finished no higher than third since the conference established divisions in 1992.
Kentucky trailed after three quarters when Calvin Dawson scored his third touchdown on a one-yard plunge, and the extra point put Louisiana-Monroe on top 34-33. But the Wildcats took the lead for good four minutes later on a 34-yard field goal by Lones Seiber.
Although the Kentucky offense has dominated this year, the defense - particularly the run defense - struggled again Saturday.
Kentucky led by 18 midway through the second quarter after an 84-yard punt return down the right sideline by Rafael Little - the seventh-longest punt return score in Kentucky history. Louisiana-Monroe (2-8, 1-4 Sun Belt) would follow with a 3-yard touchdown run by Dawson and add a 41-yard field goal by Cole Wilson to cut the Wildcats' lead to 28-20 at halftime.
Dawson then dominated the Warhawks' opening possession of the second half. On consecutive, plays he ran for 24 and 16 yards before skipping into the end zone untouched from nine yards out.
A sack by Kentucky defensive end Myron Pryor forced the Warhawks to punt from deep in their own territory, but punter Kevin Payne bobbled the snap out of the back of the end zone. That gave the Wildcats a safety and possession, which produced a 26-yard field goal by Seiber, stretching the lead back to six.
Louisiana-Monroe led early when Travis Eickman leaped into the air to block Seiber's 25-yard field goal attempt on Kentucky's first possession, and Greg James scooped it to travel 78 yards for the touchdown.
The Wildcats made the most of their opportunity on their next drive. Little took a handoff and spun through two defenders to sprint 46 yards and put Kentucky in position. Then, backup running back Alfonso Smith plowed through traffic to score from eight yards out and tie the game at 7.
Cole Wilson managed a high snap and nailed a 45-yard field goal to put the Warhawks back in front 10-7.
Kentucky regained the lead at the end of the opening quarter when Woodson found Burton wide open for a 37-yard touchdown reception. Burton also scored from 17 yards.